Let be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of that are distinguished by its subgroup . One relates this class to representations which come as base change lifts from a quasi-split unitary group over F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose -distinction, and thus exhibit the limitations of these two approaches.
Funding source: Israel Science Foundation
Award Identifier / Grant number: 756/12
Funding source: Hong Kong Institute of Educational Research, Chinese University of Hong Kong
Award Identifier / Grant number: CUHK 405213
Funding source: Israel Science Foundation
Award Identifier / Grant number: 1138/10
Funding statement: Maxim Gurevich, partially supported by the ISF grant 756/12, and ERC StG grant 637912. Arnab Mitra, partially supported by postdoctoral fellowships funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev and the Department of Mathematics, Technion.
The authors would like to thank Wee Teck Gan, Erez Lapid, Omer Offen, Dipendra Prasad, Eitan Sayag, and Jiu Kang Yu for several helpful conversations. The third author would like to thank Steven Spallone for answering his questions on parity of self-dual representations and sharing his notes on the subject matter with him. The second and the third author would like to thank the Hausdorff Institute for Mathematics (Bonn) for its warm hospitality where this project was initiated. Part of the work was done during the third author’s visit to CUHK (Hong Kong). It is a pleasure for him to thank Jiu Kang Yu for inviting him and the institute for providing an excellent work environment.
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